Tuesday, January 22, 2008

Fermat's Lost Treasure

Fermat's Last Theorem (FLT): a^n + b^n = c^n has no integer solution for n > 2.

In the geometry of one dimensional space, n=1, the stacking of line segments, FLT has integer solutions for all integers a and b.

In 2 dimensional geometry, FLT is The Pythagorean Theorem. Integer solutions are known as Pythagorean Triples. There are infinitely many. Ancient artifacts from multiple ancient cultures have been found prominently featuring Pythagorean Triples.

In higher dimensions integer solutions are known for equations with more than three factors, for example:

n=2: {3, 4, 5}
n=3: {3, 4, 5, 6}
n=4: {30, 120, 272, 315, 353}
n=5: {19, 43, 46, 47, 67, 72}
n=7: {127, 258, 266, 413, 430, 439, 525, 568}
n=8: {90, 223, 478, 524, 748, 1088, 1190, 1324, 1409}
[http://en.wikipedia.org/wiki/Pythagorean_triples#Generalizations)]

I will call this generalized FLT (GFLT). What do these numbers describe geometrically in two dimensional space, three dimensional, four, etc.?

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